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Using the Cosine Function to Find the Angle (Mathematics Lesson)

Using the Cosine Function to Find the Angle of a Right Triangle

The cosine function relates a given angle to the adjacent side and hypotenuse of a right triangle.

The angle (labelled θ) is given by the formula below:



In this formula, θ is an angle of a right triangle, the adjacent is the length of the side next to the angle and the hypotenuse is the length of longest side. cos-1 is the inverse cosine function (see Note). The image below shows what we mean:

How to Use the Cosine Function to Find the Angle of a Right Triangle

Finding the angle of a right triangle is easy when we know the adjacent and the hypotenuse.

An Example Question

What is the angle of the right triangle shown below?

Step 1
Start with the formula:
θ = cos-1 (adjacent / hypotenuse)
Don't forget: cos-1 is the inverse cosine function (it applies to everything in the brackets) and / means ÷

Step 2
Substitute the length of the adjacent and the length of the hypotenuse into the formula. In our example, the adjacent is 3 cm and the hypotenuse is 6 cm.
θ = cos-1 (3 / 6)
θ = cos-1 (3 ÷ 6)
θ = cos-1 (0.5)
θ = 60°
The angle of a right triangle with an adjacent of 3 cm and a hypotenuse of 6 cm is 60°.

Remembering the Formula

Often, the hardest part of finding the unknown angle is remembering which formula to use.

Whenever you have a right triangle where you know two sides and have to find an unknown angle...

......think trigonometry...

...............think sine, cosine or tangent...

........................think SOH CAH TOA.



Looking at the example above, we know the Adjacent and the Hypotenuse.



The two letters we are looking for are AH, which comes in the CAH in SOH CAH TOA.

This reminds us of the equation:
Cos θ = Adjacent / Hypotenuse
We want the angle, θ, not the cosine of the angle, cos θ. To do this, we need to taken the inverse cosine, cos-1 (see Note).
θ = Cos-1 (Adjacent / Hypotenuse)

A Real Example of How to Use the Cosine Function to Find the Angle of a Right Triangle

The slider below gives another example of finding the angle of a right triangle (if the hypotenuse and adjacent are known).
Interactive Widget
Here is an interactive widget to help you learn about the cosine function.
Interactive Test
  show
 


Here's a second test on finding the angle using the cosine function.
Here's a third test on finding the angle using the cosine function.
Note

What Is the Inverse Cosine Function?

The inverse cosine function is the opposite of the cosine function.

The cosine function takes in an angle, and gives the ratio of the adjacent to the hypotenuse:



The inverse cosine function, cos-1, goes the other way. It takes the ratio of the adjacent to the hypotenuse, and gives the angle:

Switch Sides, Invert the Cosine

You may see the cosine function in an equation:



To make θ the subject of the equation, take the inverse cosine of both sides.

The inverse cosine cancels out the cosine on the left hand side of the equals side, so the equation looks as below:



Comparing the two equations, the cosine has moved from one side of the equals sign to the other and has changed from cos to cos-1.

(Note: the reverse is also true. A cos-1 can be moved to the other side of the equals sign, where it becomes a cos.)

Other Inverse Trigonometric Functions

Just as the cosine function has an inverse, so do the sine and tangent functions.